Euler characteristics of categories and homotopy colimits

Abstract: The Euler characteristic is among the earliest and most elementary homotopy invariants. For a finite simplicial complex, it is the alternating sum of the numbers of simplices in each dimension. This combinatorially defined invariant has remarkable connections to geometric notions, such as genus, curvature, and area.

Euler characteristics are not only defined for simplicial complexes or manifolds, but for many other objects as well, such as posets and, more generally, categories. We propose in this talk a topological approach to Euler characteristics of categories. The idea, phrased in homological algebra, is the following. Given a category Γ\Gamma and a ring RR, we take a finite projective RΓR\Gamma-module resolution P*P_* of the constant module R̲\underline{R} (assuming such a resolution exists). The alternating sum of the modules PiP_i is the finiteness obstructiono(Γ,R)o(\Gamma,R). It is a class in the projective class group K0(RΓ)K_0(R\Gamma), which is the free abelian group on isomorphism classes of finitely generated projective RΓR\Gamma-modules modulo short exact sequences. From the finiteness obstruction we obtain the Euler characteristic respectively L2L^2-Euler characteristic , by adding the entries of the RΓR\Gamma-rank respectively the L2L^2-rank of the finiteness obstruction.

This topological approach has many advantages, several of which now follow. First of all, this approach is compatible with almost anything one would want, for example products, coproducts, covering maps, isofibrations, and homotopy colimits. It works equally well for infinite categores and finite categories. There are many examples. Classical constructions are special cases, for example, under appropriate hypotheses the functorial L2L^2-Euler characteristic of the proper orbit category for a group GG is the equivariant Euler characteristic of the classifying space for proper GG-actions. The K-theoretic Möbius inversion has Möbius-Rota inversion and Leinster’s Möbius inversion as special cases. We also obtain the classical Burnside ring congruences.

This talk will focus on our Homotopy Colimit Formula for Euler characteristics.

In certain cases, the L2L^2-Euler characteristic agrees with the groupoid cardinality of Baez-Dolan and the Euler characteristic of Leinster, and comparisons will be made.

Dendroidal weak nn-categories

Abstract:Dendroidal sets were introduced by I. Moerdijk and I. Weiss as a generalization of simplicial sets, suitable to study operads in the context of homotopy theory in MoerWeiss07a. The idea behind the notion of dendroidal sets is that in the same way as simplicial sets help us understanding categories via the nerve functor, there should be an analogous notion for studying coloured operads as generalization of categories. Much of the fundamentals of simplicial sets that relate to category theory extend to dendroidal sets. To name a few,

I. Moerdijk and I. Weiss developed the theory of inner Kan complexes in the category of dendroidal sets in MoerWeiss07b;

The purpose of this talk is to show that dendroidal sets can also be used to give a new definition of weak n-categories, and to compare the result with the corresponding classical notions in low degrees: bicategories and tricategories.

We discuss different (non)equivalent notions of regularity and exactness for 2-categories and bicategories by reviewing basic examples: the 2-category Cat of small categories, then the 2-category CatBcoopCat^{\B^{coop}} of B\B-indexed categories for a small bicategory ℬ\mathcal{B}, and the 2-category St(X)St(X) of stacks over a topological space XX. All these examples are crucial in two-dimensional topos theory: the 2-topos Cat plays a role of the topos Set in ordinary topos theory and a point of any 2-topos ℰ\mathcal{E} is defined as a 2-geometric morphismp:Cat→ℰp : Cat \to \mathcal{E}. The second example CatBcoopCat^{\B^{coop}} is a 2-topos which we denote by BℬB\mathcal{B}, and call the classifying 2-topos of a small bicategory ℬ\mathcal{B}, in an analogy with a classifying topos of small bicategory ℬ\mathcal{B} introduced in BaJu09. We use a third example of a 2-topos St(X)St(X) in order to introduce a notion of a 2-torsor over XX under a bicategory ℬ\mathcal{B}. These are homomorphisms 𝒫:ℬ→St(X)\mathcal{P} : \mathcal{B} \to St(X) such that stalks over XX of the canonical fibration of bicategories π𝒫:∫ℬ𝒫→ℬ\pi_{\mathcal{P}} : \int_{\mathcal{B}} \mathcal{P} \to \mathcal{B} given by a bicategorical version of the Grothendieck construction, are 2-filtered in the sense of Dubuc and Street. Our main result is a bicategorical version of Diaconescu's theorem which says that there exists a natural biequivalence

where the left side is a 2-category of geometric 2-morphisms from the Grothendieck 2-topos St(X)St(X) to the classifying 2-topos BℬB\mathcal{B} of ℬ\mathcal{B} and the right side is the 2-category of left ℬ\mathcal{B}-2-torsors over XX. We discuss connections of this result with a joint work of Bunge and Hermida.